Localization of Hopf bifurcations in fluid flow problems

International Journal for Numerical Methods in Fluids

Volumn 24, Issue 11, 1997, Pages 1185-1210

  Fortin, A ^a   Jardak, M ^a   Gervais, J J ^b   Pierre, R ^b  

^a ÉCOLE POLYTECHNIQUE DE MONTRÉAL   (Canada)

^b UNIVERSITÉ LAVAL   (Canada)

Author keywords

Direct simulation; Eigenproblem; Hopf bifurcation; Hydrodynamic stability; Navier Stokes equations

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; FLOW OF FLUIDS; NAVIER STOKES EQUATIONS; REYNOLDS NUMBER; STABILITY;

DIRECT SIMULATION; HOPF BIFURCATIONS; JACOBIAN MATRIX; LOCALIZATION;

BIFURCATION (MATHEMATICS);

BIFURCATIONS; FLUID FLOW; NAVIER-STOKES EQUATIONS;

EID: 0031171280     PISSN: 02712091     EISSN: None     Source Type: Journal    
DOI: 10.1002/(sici)1097-0363(19970615)24:11<1185::aid-fld535>3.0.co;2-x     Document Type: Article

References (33)

1. A. Fortin, M. Jardak, J. J. Gervais and R. Pierre, 'Old and new results on the two-dimensional Poiseuille flow', J. Comput. Phys., 115, 455-469 (1994).
2. A. Fortin, M. Fortin and J. J. Gervais, 'Complex transition to chaotic flow in a periodic array of cylinders', Theor. Comput. Fluid Dyn., 3, 79-93 (1991).
3. J. W. Goodrich, K. Gustafson and K. Halasi, 'Hopf bifurcation in the driven cavity', J. Comput. Phys., 90, 219-261 (1990).
4. K. Gustafson and K. Halasi, 'Vortex dynamics of cavity flows', J. Comput. Phys., 64, 279-319 (1986).
5. K. Gustafson and K. Halasi, 'Cavity flow dynamics at higher Reynolds number and higher aspect ratio', J. Comput. Phys., 70, 271-283 (1987)
6. P. M. Gresho, D. K. Gartling, J. R. Torczynsi, K. A. Cliffe, T. J. Garratt, A. Spence and J. W. Goodrich, 'Is the steady viscous incompressible two-dimensional flow over a backward-facing step stable at Re = 800?' Int. j. numer. methods fluids, 17, 501-541 (1993).
7. J. J. Gervais, D. Lemelin and R. Pierre, 'Some experiments with stability analysis of discrete incompressible flows in the lid-driven cavity', Int. j. numer. methods fluids, 24, 477-492 (1997).
8. K. Winters, 'Bifurcation and stability, a computational approach', Comput. Phys. Commun., 65, 299-309 (1991).
9. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983.
10. M. Jardak, Ph.D. Thesis, École Polytechnique de Montréal, 1995.
11. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, New York, 1991.
12. A. Georgescu, Hydrodynamic Stability Theory, Martinus Nijhoff, 1985.
13. A. Jennings, Matrix Computation for Engineers and Scientists, Wiley, London, 1977.
14. W. J. Stewart and A. Jennings, 'A simultaneous iteration algorithm for real matrices', ACM Trans. Softw., 7, 230-242 (1981).
15. K. L. Bathe and E. L. Wilson, Numerical Methods in Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1976.
16. M. Fortin and A. Fortin, 'Experiments with several elements for viscous incompressible flows', Int. j. numer. methods fluids, 5, 911-928 (1985).
17. M. Fortin, 'Old and new finite elements for incompressible flows', Int. j. numer. methods fluids, 1, 347-364 (1981).
18. K. A. Cliffe, T. J. Garratt and A. Spence, 'Eigenvalues of discretized Navier-Stokes equation with application to the detection of Hopf bifurcations', Adv. Comput. Math., 1, 337-356 (1993).
19. R. Neubert, 'Predictor-corrector techniques for detecting Hopf bifurcation points', Bifurc. Chaos Appl. Sci. Eng., 3 1311-1318 (1993).
20. Y. Saad, 'Chebychev acceleration techniques for solving non-symmetric eigenvalue problems', Math. Comput., 42, (1984).
21. D. C. Sorenson, 'Implicit application of polynomial filters in a K-step Amoldi method', SIAM J. Matrix Anal. Appl., 13, 357-385 (1992).
22. I. S. Duff and J. A. Scott, 'Computing selected eigenvalues of sparse unsymmetric matrices using subspace iteration', ACM Trans. Math. Softw., 19, 137-159 (1991).
23. K. A. Cliffe, T. J. Garratt and A. Spence, 'Eigenvalues of block matrices arising from problems in fluid mechanics', SIAM J. Matrix Anal. Appl., 15, 1310-1318 (1994).
24. A. Griewank and G. Reddien, 'The calculation of Hopf points by a direct method', IMA J. Numer. Anal., 3, 295-304 (1983).
25. M. Fortin and A. Fortin, 'A generalizatin of Uzawa algorithm for the solution of the Navier-Stokes equations', Commun. Appl. Numer. Methods, 1, 205-208 (1985).
26. S. A. Orszag, 'Accurate solution of the Orr-Sommerfeld stability equation', J. Fluid Mech., 50, 689-703 (1971).
27. K. Gustafson, 'Theory and computation of periodic solutions of autonomous partial differential equation boundary value problems, with application to the driven cavity', Math. Comput. Model, 22, 57-75 (1995).
28. C. H. Bruneau and C. Jouron, 'Un nouveau schéma décentré pour le problème de cavité entraînée', C. R. Acad. Sci. Paris, 307, 359-362 (1988).
29. J. Shen, 'Hopf bifurcation of the unsteady regularized driven cavity flow', J. Comput. Phys., 95, 228-245 (1991).
30. L. Kaiktsis, G. E. Karniadakis and S. Orszag, 'Onset of three-dimensionality, equilibria and early transition in the flow over a backward facing step', J. Fluid Mech., 231, 501-528 (1991).
31. N. K. Ghaddar, K. Z. Korczak, B. B. Mikic and A. T. Patera, 'Numerical investigation of incompressible flow in grooved channel, Part 1. Stability and self-sustained oscillations', J. Fluid Mech., 163, 99-127 (1986).
32. A. Fortin, On the imposition of a flowrate by an augmented Lagrangian method', Commun. Appl. Numer. Methods, 4, 835-841 (1988).
33. D. K. Gartling, 'A test problem for outflow boundary conditions', Int. j. numer. methods fluids, 11, 953-967 (1990).